Emergence of charge-$4e$ superconductivity from 2D nematic superconductors

TL;DR

This work investigates how vestigial charge-$4e$ order can emerge from a two-dimensional nematic superconductor by proposing a honeycomb-bond generalized XY model that couples uniform and nematic superconducting components. Using a combination of Landau theory, dual vortex analysis, and large-scale Monte Carlo simulations, it maps the finite-temperature phase diagram and reveals an intermediate quasi-nematic phase stabilized by domain-wall excitations, in addition to a charge-$4e$-ordered phase. The phase transitions are governed by the proliferation of distinct topological defects—half-SC vortices, $(\tfrac{1}{2},\tfrac{1}{2})$ vortices, integer nematic vortices, and domain walls—with domain walls playing a crucial role in stabilizing vestigial $4e$ order. A tricritical point $J_3^*$ separates different melting paths, providing a concrete microscopic mechanism for charge-$4e$ vestigial order and highlighting the importance of domain-wall physics alongside vortices in melting multi-component superconductors.

Abstract

Charge-$4e$ superconductivity is an exotic state of matter that may emerge as a vestigial order from a charge-$2e$ superconductor with multicomponent superconducting order parameters. Showing its emergence in a microscopic model from numerically-exact large-scale computations has been rare so far. Here, we propose a microscopic lattice model with a nematic superconducting ground state and show that it supports a rich set of vestigial phases at elevated temperature, including a charge-$4e$ phase and a quasi-long-range nematic phase, by performing large-scale Monte Carlo simulations. Combining theoretical analysis with Monte Carlo simulations, we uncover the nature of these phases and show that the phase transitions are governed by the proliferation of distinct topological defects: half superconducting vortices, $(\tfrac{1}{2},\tfrac{1}{2})$ vortices, integer nematic vortices, and domain-wall excitations. In particular, we demonstrate that domain-wall proliferation is crucial for the quasi-nematic phase and should be carefully accounted for in phase transitions associated with vestigial charge-$4e$ order.

Figures (8)

• Figure 1: (a) Honeycomb lattice with fields $\theta_i$ residing on the bonds and Josephson couplings $J_1$, $J_1'$, and $J_3$. (b) Ground-state configurations for different ratios of $J_1'/J_1$. (c) Phase diagram of the generalized XY model with $J_1=1$ and $J_1'=3.2$. The yellow star $(J_3^*,T^*)$ marks the tricritical point. Phase I: nematic SC state with charge-$2e$ QLRO and nematic LRO; Phase II: restoration of discrete rotational symmetry leading to a quasi-long-range nematic phase; Phase III: charge-$4e$ QLRO without nematicity; Phase IV: disordered phase. Colored points denote transition points obtained from different physical observables Note1.
• Figure 2: Demonstration of different domains with a length of 4 units. The values of $(\theta^x,\theta^y,\theta^z)$ are $(0,0,\pi)$ in the upper part, and (a) $(\pi,\pi,0)$, (b) $(0,\pi,0)$, and (c) $(\pi,0,\pi)$ below the domain wall (the shaded region). The energy cost of the domain wall increases with its length and is proportional to $6J_3$, $4J_3$, and $2J_3$ for cases (a), (b), and (c), respectively.
• Figure 3: The plot of Binder cumulants $U(L)$ and the RG-invariant ratio $R(L)$ for the different orders at $J_3=0.4$. The insets depict a close-up view of the transition points. (a)-(b) correspond to the charge-$4e$ order, where the dashed line denotes the transition temperature $T_{4e}=2.849$. (c)-(d) represent the nematic order, where the dashed lines denote the transition temperatures $T_{\text{nem}}=1.87$ and $T_{\text{nem}}'\simeq T_{\text{dw}}=1.59$. (e)-(f) show the charge-$2e$ order, where the dashed line denotes the transition temperature $T_{\text{dw}}=1.59$.
• Figure 4: (a). The size-dependence of spin stiffness $\rho$ for $J_3=0.4$. The straight line is $\rho=8T/\pi$. The inset figure is the finite-size extrapolation of the critical temperature $T^*(L)$ using $T^*(L)=T_{4e}(\infty)+\frac{a}{\ln^2(bL)}$, where $T^*(L)$ is the crossing point with line $\rho=8 T/\pi$. (b). The size-dependence of spin stiffness $\rho$ for $J_3=1.2$. The two straight lines are $\rho=8 T/\pi$ and $2 T /\pi$. The vertical dashed line represents the critical temperature extracted from $U_{4e}$, indicating a stiffness jump lies between $8 T/ \pi$ and $2 T/\pi$. The inset figure is the finite-size extrapolation of the critical temperature $T^*(L)$ using $T^*(L)=T_{4e}(\infty)+\frac{a}{L^x}$, where $T^*(L)$ is the crossing point with line $\rho=2 T/\pi$.
• Figure 5: The heat capacity for (a) $J_3=0.4$ and (b) $J_3=1.2$. The number of bumps indicates the number of transitions, which is three and two, respectively.